Is it possible to classify the subject of a conversation?
This is a difficult problem.
First, how do you define 'subject'? Do you have a (closed) lists of labels you want to assign? What about subjects that overlap, or don't occur in your list? What even is a subject? This is a non-trivial issue.
Second, and this is even harder, how do you want to recognise subjects? A simple solution could be using a list of associated keywords, but this is problematic as many words have multiple meanings, and words are not really a good indicator of a conversation topic in the first place.
Instead of jumping to an implementation method, be clear about how you want to tackle these two items first. Start by annotating a conversation transcript by hand. You will then get a feeling for the problems and possible solutions. After you have done this, you can think about how to get a machine to do it efficiently.
UPDATE: For a scheme to annotate the functions of lines within a conversation have a look at Francis &amp; Hunston (1992) Analysing Everyday Conversation.  In Coulthard, M. (ed.) &quot;Advances in Spoken Discourse Analysis&quot;. London: Routledge. pp.123-161.  This is more oriented towards linguistics, but might give you some ideas on how to proceed.